Can Faceless Void Use Basher?

Yes it is. You cannot buy basher on void, troll, slardar and spirit breaker.

Does basher work on illusions?

Yes and No, Illusion benefits from a lots of passive abilities may they be granted by items or skills but not all. You can not benefit from bashes on illusion (Bash from basher, Abyssal blade, or Barathrum's bash for example).

Does Basher stack with Troll?

So basher does not stack with any of ability bashes (troll, SB, slardar or void).